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MAC 2103

MAC 2103. Module 7 Euclidean Vector Spaces II. Learning Objectives. Upon completing this module, you should be able to: Determine if a linear operator in ℜ n is one-to-one. Find the inverse of a linear operator in ℜ n .

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MAC 2103

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  1. MAC 2103 Module 7 Euclidean Vector Spaces II

  2. Learning Objectives Upon completing this module, you should be able to: • Determine if a linear operator in ℜn is one-to-one. • Find the inverse of a linear operator in ℜn . • Use the images of the standard basis vectors to find a standard matrix in ℜn . • Find the polynomial q=T(p) in P1 corresponding to the transformation T on any polynomials in P1. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  3. Euclidean Vector Spaces II There are two major topics in this module: Properties of Linear Transformations from ℜn to ℜm Linear Transformations and Polynomials http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.09

  4. What are the Important Properties of Linear Transformations? A transformation T: ℜn → ℜm is linear if both of the following relationships hold for all vectorsu and vin ℜn and for every scalar s: (See Theorem 4.3.2) T(u + v) = T(u) + T(v) b) T(su) = sT(u) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  5. What are the Important Properties of Linear Transformations? (Cont.) It follows that: T(-v) = T[(-1)v] = (-1)T(v) = - T(v), T(u - v) = T[u + (-1) v] = T(u) + (-1)T(v) = T(u) - T(v), T(0) = T(0v) = 0T(v) = 0, since 0v = 0; and http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  6. How to Determine if a Linear Operator in ℜn is one-to-one? Example: Find the standard matrix for the linear operator defined by the equations and determine whether the operator is one-to-one? (a) Solution: Since det(A) = 0, the matrix is not invertible. Thus, the linear operator in this case is not one-to-one. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  7. How to Determine if a Linear Operator in ℜn is one-to-one? (Cont.) Example: Find the standard matrix for the linear operator defined by the equations and determine whether the operator is one-to-one? (b) Solution: Since det(A) ≠ 0, the matrix is invertible. Thus, the linear operator in this case is one-to-one. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  8. How to Find the Inverse of the Linear Operator in ℜn ? Example: Find the inverse of the operator if the operator is one-to-one? Solution: From the previous slide, we have checked that the linear operator is one-to-one. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  9. How to Find the Inverse of the Linear Operator in ℜn ? (Cont.) Thus, Check: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  10. What are the Standard Basis Vectorsin ℜn ? The standard basis vectors in ℜn are the columns of In (the identity matrix in ℜn). We have represented the standard basis vectors in ℜ3 as i, j , and k; In order to extend the notations to ℜn, we can represent them as e1, e2, e3 (note that the hat notation used below is generally reserved to denote unit vectors) as follows: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  11. What are Standard Basis Vectorsin ℜn? (Cont.) As mentioned previously, the standard basis vectors in ℜn are the columns of the In, we can represent them in ℜn, as e1, e2, … , en as follows: Thus, http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  12. How to Find the Standard Matrix from the Images of the Standard Basis Vectors in ℜn? Note: If the linear transformation is represented by T: ℜn → ℜm or TA: ℜn → ℜm; the matrix A = [aij] is called the standard matrix for the linear transformation, and T is called multiplication by A. Now we can use the images of the standard basis vectors to find the standard matrix. As we have learned in module 6, if A is the standard matrix for T: ℜn → ℜm, then A = [TA] = [T]. Thus, http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  13. How to Find the Standard Matrix from the Images of the Standard Basis Vectors in ℜn? (Cont.) • Example: Find the standard matrix for T: ℜ3 → ℜ3from the images of the standard basis vectors, where T: ℜ3 → ℜ3reflects a vector about the xz-plane and then contracts that vector by a factor of 1/2. • Solution: We want to find the standard matrix from the images of the standard basis vectors, • z • y • e1 • x http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  14. How to Find the Standard Matrix from the Images of the Standard Basis Vectors in ℜn? (Cont.) z e2 y x http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  15. How to Find the Standard Matrix from the Images of the Standard Basis Vectors in ℜn? (Cont.) z e3 y x http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  16. How to Find the Standard Matrix from the Images of the Standard Basis Vectors in ℜn? (Cont.) • The standard matrix is: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  17. How to Find the Polynomial q=T(p) in P1 Corresponding to the Transformation T on any Polynomials in P1? • Example: What is the corresponding polynomial q=T(p) on polynomials of degree ≤ 1, P1, for the multiplying matrix A: • Solution: First of all, let A be the multiplying matrix for the transformation T. • T is a linear operator on P1 for which the domain is P1 and the codomain is P1. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  18. How to Find the Polynomial q=T(p) in P1 Corresponding to the Transformation T on any Polynomials in P1?(Cont.) Thus, if p is a polynomial of degree ≤ 1 and p(x) = ax1 + bx0 is a linear combination with real-valued coefficients of x1 = x and x0 = 1, which are linearly independent functions (we will discuss this in module 8), for some real numbers a and b. Then, A multiplies the vector of coefficients of p(x). and the corresponding transformation on p(x) is as follows: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  19. What have we learned? We have learned to: • Determine if a linear operator in ℜn is one-to-one. • Find the inverse of a linear operator in ℜn . • Use the images of the standard basis vectors to find a standard matrix in ℜn . • Find the polynomial q=T(p) in P1 corresponding to the transformation T on any polynomials in P1. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

  20. Credit Some of these slides have been adapted/modified in part/whole from the following textbook: • Anton, Howard: Elementary Linear Algebra with Applications, 9th Edition http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.F09

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